1-1_Set_Terminology_and_Notation

Logic and Mathematics

• Mathematics relies on deductive reasoning.

• Conclusions are logically derived from statements accepted as true.

• Proofs are essential to confirm that conclusions are universally true.

• Unlike natural/social sciences, mathematics demands absolute certainty that statements are true in all cases.

• Validity is confirmed through formal proof rather than checking individual examples.

Set Terminology and Notation

Importance of Definitions

• Mathematical definitions differ from ordinary language.

• They possess precise, fixed meanings to avoid ambiguity.

• Example: An integer is odd means it meets a specific condition, not that it is unusual.

Notation Importance

• Correct notation ensures clarity, precision, and prevents errors.

• Mathematics is a structured language that requires proper notation for clear understanding.

Definitions of Sets

• Definition 1.1.1: A set is a collection of objects, called elements.

  • If x is an element of set A, write x ∈ A.

  • If x is not an element of set A, write x ∉ A.

  • The empty set is denoted by ∅.

• Remark 1.1.2: Methods to describe sets include:

  • Roster Method: Listing elements when few are present (e.g., A = {6, 7, 8, 9}).

  • Description Method: Describing by properties using set-builder notation (e.g., A = {x | 5 < x < 10}).

Common Notations for Sets

• Notation 1.1.3: Frequently used sets have special notations:

  • N: Set of positive integers (natural numbers).

  • Z: Set of all integers.

  • Q: Set of all rational numbers.

  • R: Set of all real numbers.

  • C: Set of all complex numbers.

  • Q⁺, R⁺: Sets of all positive rational and real numbers respectively.

Examples and Applications

Example 1.1.4: Description of Sets

• A: {x ∈ N | x ≤ 6} => A = {1, 2, 3, 4, 5, 6}.

• B: {x ∈ Q | 1 ≤ x ≤ 6} => Elements include 3, 7, and 13.

• C: {x ∈ Q | x² ≠ 0} => C = Q , {0}.

• D: {2, -1, 0, 1, 2} can be rewritten as {x ∈ Z | -1 ≤ x ≤ 2}.

Set Cardinality and Subsets

• Definition 1.1.5: Cardinality of a set S, denoted |S|, is the number of elements in S.

  • Finite set if |S| = n, infinite otherwise.

• Definition 1.1.6: Subset definitions:

  • A is a subset of B (A ⊆ B) if all elements of A are in B.

  • A and B are equal if A ⊆ B and B ⊆ A.

  • A is a proper subset of B (A ⊂ B) if A ⊆ B but A ≠ B.

Example 1.1.7: Subset Relationships

• A = {1, 2, 3}, B = {1, 2, 3, 4} => A ⊂ B, A = C where C is a permutation of A.

• D = {2, 4, 6} is not a subset of B because 6 ∉ B.

Interval Notations for Subsets

• Notation 1.1.8: Subsets of R can be denoted as intervals:

  • Open interval (a, b): {x ∈ R | a < x < b}.

  • Closed interval [a, b]: {x ∈ R | a ≤ x ≤ b}.

  • Half-open intervals and half-closed intervals representing different bounds.

Example 1.1.9: Intervals Example

• A = {x ∈ R | x² < 4} = (-2, 2).

• B = {x ∈ R | x < 2} = (-∞, 2).

• Thus, A ⊆ B.

Power Sets and Set Operations

• Definition 1.1.11: The power set P(A) is the set of all subsets of A.

• Example 1.1.12: For A = {1, 2, 3}, P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} with |P(A)| = 8.

Set Operations

Operations Overview

• Definition 1.1.13 for two sets A and B:

  • Intersection: A ∩ B = {x | x ∈ A and x ∈ B}.

  • Union: A ∪ B = {x | x ∈ A or x ∈ B} (inclusive).

  • Difference: A \ B = {x | x ∈ A and x ∉ B}.

  • Complement: U \ A (where U is the universal set).

Venn Diagrams

• Use Venn diagrams to visually represent set operations: intersection, union, difference, complement.

Example 1.1.14: Set Examples

• Consider sets [3, 6] and [4, 8):

  • Intersection: [3, 6] ∩ [4, 8) = [4, 6].

  • Union: [3, 6] ∪ [4, 8) = [3, 8).

  • Difference: [3, 6] \ [4, 8) = [3, 4).

Ordered Pairs and Cartesian Product

• Definition 1.1.16: The Cartesian Product A × B consists of all ordered pairs {(a, b) | a ∈ A, b ∈ B}.

• Example 1.1.17: Let A = {a, b}, B = {1, 2, 3, 4}:

  • A × B = {(a, 1), (a, 2), (a, 3), (a, 4), (b, 1), (b, 2), (b, 3), (b, 4)}.